The Annals of Probability

Geometric characterization of intermittency in the parabolic Anderson model

Jürgen Gärtner, Wolfgang König, and Stanislav Molchanov

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Abstract

We consider the parabolic Anderson problem tuu+ξ(x)u on ℝ+×ℤd with localized initial condition u(0, x)=δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑xu(t, x) comes from a slowly increasing number of “islands” which are located far from each other. These “islands” are local regions of those high exceedances of the field ξ in a box of side length 2t log2t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.

Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 439-499.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1175287751

Digital Object Identifier
doi:10.1214/009117906000000764

Mathematical Reviews number (MathSciNet)
MR2308585

Zentralblatt MATH identifier
1126.60091

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F10: Large deviations 35B40: Asymptotic behavior of solutions

Keywords
Parabolic Anderson problem intermittency random environment quenched asymptotics heat equation with random potential

Citation

Gärtner, Jürgen; König, Wolfgang; Molchanov, Stanislav. Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007), no. 2, 439--499. doi:10.1214/009117906000000764. https://projecteuclid.org/euclid.aop/1175287751


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